Method for carrying out integer approximation of transform coefficients, and coder and decoder

ABSTRACT

A limited value range is defined for obtaining approximated integer cosine transform coefficients.  
     The transform coefficients of the base vectors for the submatrices are selected by taking into account the orthogonality condition so that the sum of their squares yields the square of the constant component coefficient. The coefficients of the variable components are derived from these coefficients.  
     These measures yield the advantage that in quantization and normalization, a uniform normalization and quantization factor may be used for all coefficients.

BACKGROUND INFORMATION

[0001] Integer approximations of DCT (Discrete Cosine Transform) areused in frequency range coding of moving image sequences in which a pureinteger arithmetic is required. Use of integer arithmetic preventsdivergence between the coder and decoder due to different computationaccuracies.

[0002] A 4×4 block size integer cosine transform is proposed for thetest model of video coding standard H.26L. A concept which is used fortransformation of block sizes linked to motion compensation is proposedin [1]. To do so, integer cosine transforms of an 8×8 and 16×16 blocksize are also necessary, the coefficients being selectable only in alimited range because of the limited numerical range available, e.g., 32bit. Orthogonal integer cosine transforms of an 8×8 block size have beenpublished in [2], for example.

[0003] There are various approaches for approximation of the DCT by aninteger cosine transform. A method in which coefficients havingdifferent absolute values in DCT are replaced by integers is describedin [3] and [4].

[0004] A method of approximation of DCT using the Hadamard transform isdescribed in [5]. The DCT may be generated by multiplying the Hadamardtransform times an orthogonal conversion matrix. By replacing the realconversion matrix with an orthogonal matrix having integer coefficients,an integer approximation of the DCT is generated.

ADVANTAGES OF THE INVENTION

[0005] The method according to the steps of Patent claim 1 and thedependent claims supplies integer transform matrices, e.g., for blocksizes of 8×8 and 16×16, having approximately the properties of DCT forfrequency range coding of pixel blocks for moving image sequences, thusmaking it possible to increase coding efficiency.

[0006] Coding may be performed with the measures of the presentinvention in pure integer arithmetic. Through the choice of thetransform coefficients according to the present invention, a maximum ofdifferent possible integer coefficients representing a goodapproximation to the non-integer DCT coefficients may be used in alimited numerical range, preferably 32 bit. The integer coefficients ofthe transform are selected to be less than 2⁹ with a view to thequantization and normalization of the transformed coefficients in theabove numerical range. Because of integer arithmetic, these may beimplemented by tabular values.

[0007] It is especially advantageous according to the present inventionthat the coefficients of all base vectors have the same norm, whichmeans that in quantization and normalization, a uniform normalizationand quantization factor may be used. However, in the methods accordingto [3] and [4], the even and odd base functions of the stated integertransforms have a different norm, so that these matrices do not fulfillthe requirement of orthogonality. No methods which would result inorthogonal integer approximations of DCT, e.g., of the block size 16×16,are given in [5].

DESCRIPTION OF EXEMPLARY EMBODIMENTS

[0008] Before explaining the actual invention, first a few prerequisitesshall be explained.

[0009] An orthogonal integer approximation of the N×N DCT must meet theconditions

T _(N) ·T _(N) ^(T) =k _(N) ·I _(N)  (1)

T _(N) ^(T) ·T _(N) =k _(N) ·I _(N)  (2)

[0010] where I_(N) is the unit matrix of the size N×N and k_(N) is aninteger constant.

[0011] The orthogonal integer transform of the block size 8×8 accordingto the present invention has the lowest maximum approximation errorwithin the numerical range of the coefficients of 2⁷

e _(max)=max{e(t ₀),e(t ₁), . . . , e(t ₇)},  (3)

[0012] where $\begin{matrix}{{{e\left( t_{m} \right)} = {{1 - \frac{t_{{ICT},m}}{\sqrt{k_{8}} \cdot t_{{DCT},m}}}}},} & (4)\end{matrix}$

[0013] where t_(DCT,m) and t_(ICT,m), where m=0, . . . , 7, denote thecoefficients of DCT and ICT (integer cosine transform) respectively. Sofar, no suggestions regarding integer approximations of DCT of the blocksize 16×16 are found in the literature, i.e., transforms that meet theconditions according to equations (1) and (2) for N=16. The integertransform according to the equation meets these conditions so that incontrast with the approach mentioned in [4], all the base vectors havethe same norm. This means an advantage in particular in quantization andnormalization of the transformed coefficients, because a uniformnormalization and quantization factor may be used for all thecoefficients.

[0014] Integer approximation T_(s) of the block size 8×8 is generated byreplacing the seven different real absolute values of the coefficientsof the 8×8 DCT by seven integer values, taking into account theorthogonality conditions (1) and (2) for N=8.$T_{{ICT},8} = {\begin{pmatrix}17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 \\24 & 20 & 12 & 6 & {- 6} & {- 12} & {- 20} & {- 24} \\23 & 7 & {- 7} & {- 23} & {- 23} & {- 7} & 7 & 23 \\20 & {- 6} & {- 24} & {- 12} & 12 & 24 & 6 & {- 20} \\17 & {- 17} & {- 17} & 17 & 17 & {- 17} & {- 17} & 17 \\12 & {- 24} & 6 & 20 & {- 20} & {- 6} & 24 & {- 12} \\7 & {- 23} & 23 & {- 7} & {- 7} & 23 & {- 23} & 7 \\6 & {- 12} & 20 & {- 24} & 24 & {- 20} & 12 & {- 6}\end{pmatrix}.}$

[0015] According to [5], the DCT may be generated with the help of aconversion matrix of the size 16×16 from Hadamard transform H_(BRO) ofthe size 16×16. BRO here stands for bit-reversed order of the basevectors of the Hadamard transform, which is known from [6], for example.The real conversion matrix is replaced by an integer orthogonalconversion matrix C₁₆ so that

T _(16,BRO) =C ₁₆ ·H _(BRO).  (5)

[0016] C₁₆ is a sparse matrix having a block diagonal structure,$\begin{matrix}{C_{16} = \begin{pmatrix}{\hat{C}}_{1} & \quad & \quad & \quad & \quad \\\quad & {\hat{C}}_{1} & \quad & 0 & \quad \\\quad & \quad & {\hat{C}}_{2} & \quad & \quad \\\quad & 0 & \quad & {\hat{C}}_{4} & \quad \\\quad & \quad & \quad & \quad & {\hat{C}}_{8}\end{pmatrix}} & (6)\end{matrix}$

[0017] where

[0018] Ĉ₁ denotes an individual integer coefficient, Ĉ_(n) is an integerorthogonal N×N matrix. For the submatrices of matrix C_(n) theconstruction specification follows $\begin{matrix}{{\hat{C}}_{2n} = {\begin{pmatrix}B_{1,n} & B_{2,n} \\{{- J_{n}}B_{2,n}J} & {J_{n}B_{1,n}J_{n}}\end{pmatrix}.}} & (7)\end{matrix}$

[0019] where

[0020] J_(n) is the transposed standard matrix, e.g.,${J_{2} = \begin{pmatrix}0 & 1 \\1 & 0\end{pmatrix}},$

[0021] and B_(1,n) and B_(2,n) describe integer N×N matrices whosecoefficients approximate the properties of the real conversation matrixas closely as possible.

[0022] For Ĉ₁=17, various integer orthogonal approximations of theconversion matrix may be created and thus orthogonal integer cosinetransforms of the block size 16×16 where k₁₆=16×17² are determined.Thus, each coefficient of the constant base vectors—constantcomponent—of these integer cosine transforms (ICTs) has a value of 17.The coefficients of the submatrices for the variable components may bederived from this value 17 by the fact that squaring 17 yields the sumof the squared coefficient values of each row of the submatrices. Forthe efficients of the lowest variable component, this yields absolutevalues 15 and 8 because 17²=15²+8². The values of the coefficients forthe second lowest variable component are derived accordingly, resultingin, e.g., 12, 9, and 8, and for the third lowest variable component, ine.g., 13, 10, 4, and 2. For the sake of illustration, one of thepossible solutions for the matrix C₁₆ and thus also for T_(ICT,16) isgiven below. Matrices Ĉ_(n) may be selected as follows: $\begin{matrix}{{\hat{C}}_{1} = 17} \\{{\hat{C}}_{2} = \begin{pmatrix}15 & 8 \\{- 8} & 15\end{pmatrix}} \\{{\hat{C}}_{4} = \begin{pmatrix}12 & 0 & 9 & 8 \\0 & 12 & {- 8} & 9 \\{- 9} & 8 & 12 & 0 \\{- 8} & {- 9} & 0 & 12\end{pmatrix}} \\{{\hat{C}}_{8} = \begin{pmatrix}13 & 0 & 0 & {- 2} & 10 & 0 & 0 & 4 \\0 & 13 & 2 & 0 & 0 & 10 & {- 4} & 0 \\0 & {- 2} & 13 & 0 & 0 & 4 & 10 & 0 \\2 & 0 & 0 & 13 & {- 4} & 0 & 0 & 10 \\{- 10} & 0 & 0 & 4 & 13 & 0 & 0 & 2 \\0 & {- 10} & {- 4} & 0 & 0 & 13 & {- 2} & 0 \\0 & 4 & {- 10} & 0 & 0 & 2 & 13 & 0 \\{- 4} & 0 & 0 & {- 10} & {- 2} & 0 & 0 & 13\end{pmatrix}}\end{matrix}$

[0023] After appropriately resorting the row vectors of T_(16,BRO), thisthen results in the orthogonal integer transform matrix$T_{16} = {\begin{pmatrix}17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & | & \ldots \\25 & 21 & 25 & 21 & 9 & {- 3} & 9 & {- 3} & | & \ldots \\29 & 13 & 11 & {- 5} & 5 & {- 11} & {- 13} & {- 29} & | & \ldots \\9 & {- 3} & 9 & {- 3} & {- 25} & {- 21} & {- 25} & {- 21} & | & \ldots \\23 & 7 & {- 7} & {- 23} & {- 23} & {- 7} & 7 & 23 & | & \ldots \\25 & 21 & {- 25} & {- 21} & {- 9} & 3 & 9 & {- 3} & | & \ldots \\11 & {- 5} & {- 29} & {- 13} & 13 & 29 & 5 & {- 11} & | & \ldots \\9 & {- 3} & {- 9} & 3 & 25 & 21 & {- 25} & {- 21} & | & \ldots \\17 & {- 17} & {- 17} & 17 & 17 & {- 17} & {- 17} & 17 & | & \ldots \\21 & {- 25} & {- 21} & 25 & {- 3} & {- 9} & 3 & 9 & | & \ldots \\13 & {- 29} & 5 & 11 & {- 11} & {- 5} & 29 & {- 13} & | & \ldots \\{- 3} & {- 9} & 3 & 9 & {- 21} & 25 & 21 & {- 25} & | & \ldots \\7 & {- 23} & 23 & {- 7} & {- 7} & 23 & {- 23} & 7 & | & \ldots \\21 & {- 25} & 21 & {- 25} & 3 & 9 & 3 & 9 & | & \ldots \\{- 5} & {- 11} & 13 & {- 29} & 29 & {- 13} & 11 & 5 & | & \ldots \\{- 3} & {- 9} & {- 3} & {- 9} & 21 & {- 25} & 21 & {- 25} & | & \ldots\end{pmatrix}.}$

[0024] The row vectors are continued symmetrically as even and oddvalues according to the DCT.

[0025] Another possible orthogonal 16×16 integer transform matrix hasthe following form: $T_{16} = \begin{bmatrix}17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 & 17 \\21 & 21 & 21 & 21 & 15 & 7 & 15 & 7 & {- 7} & {- 15} & {- 7} & {- 15} & {- 21} & {- 21} & {- 21} & {- 21} \\24 & 20 & 12 & 6 & {- 6} & {- 12} & {- 20} & {- 24} & {- 24} & {- 20} & {- 12} & {- 6} & 6 & 12 & 20 & 24 \\15 & 7 & 15 & 7 & {- 21} & {- 21} & {- 21} & {- 21} & 21 & 21 & 21 & 21 & {- 7} & {- 15} & {- 7} & {- 15} \\23 & 7 & {- 7} & {- 23} & {- 23} & {- 7} & 7 & 23 & 23 & 7 & {- 7} & {- 23} & {- 23} & {- 7} & 7 & 23 \\21 & 21 & {- 21} & {- 21} & {- 15} & {- 7} & 15 & 7 & {- 7} & {- 15} & 7 & 15 & 21 & 21 & {- 21} & {- 21} \\20 & {- 6} & {- 24} & {- 12} & 12 & 24 & 6 & {- 20} & {- 20} & 6 & 24 & 12 & {- 12} & {- 24} & {- 6} & 20 \\15 & 7 & {- 15} & {- 7} & 21 & 21 & {- 21} & {- 21} & 21 & 21 & {- 21} & {- 21} & 7 & 15 & {- 7} & {- 15} \\17 & {- 17} & {- 17} & 17 & 17 & {- 17} & {- 17} & 17 & 17 & {- 17} & {- 17} & 17 & 17 & {- 17} & {- 17} & 17 \\21 & {- 21} & {- 21} & 21 & 7 & {- 15} & {- 7} & 15 & {- 15} & 7 & 15 & {- 7} & {- 21} & 21 & 21 & {- 21} \\12 & {- 24} & 6 & 20 & {- 20} & {- 6} & 24 & {- 12} & {- 12} & 24 & {- 6} & {- 20} & 20 & 6 & {- 24} & 12 \\7 & {- 15} & {- 7} & 15 & {- 21} & 21 & 21 & {- 21} & 21 & {- 21} & {- 21} & 21 & {- 15} & 7 & 15 & {- 7} \\7 & {- 23} & 23 & {- 7} & {- 7} & 23 & {- 23} & 7 & 7 & {- 23} & 23 & {- 7} & {- 7} & 23 & {- 23} & 7 \\21 & {- 21} & 21 & {- 21} & {- 7} & 15 & {- 7} & 15 & {- 15} & 7 & {- 15} & 7 & 21 & {- 21} & 21 & {- 21} \\6 & {- 12} & 20 & {- 24} & 24 & {- 20} & 12 & {- 6} & {- 6} & 12 & {- 20} & 24 & {- 24} & 20 & {- 12} & 6 \\7 & {- 15} & 7 & {- 15} & 21 & {- 21} & 21 & {- 21} & 21 & {- 21} & 21 & {- 21} & 15 & {- 7} & 15 & {- 7}\end{bmatrix}$

[0026] It is constructed as follows:

[0027] The even rows 0:2:14 correspond to the even rows of T_(ICT,8)described above, the continuation of row length 8 to 16 beingaccomplished by mirroring of the coefficients. The odd rows 1:2:15 areobtained as follows:

[0028] Ordered Hadamard matrix: $H_{16} = \begin{bmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 \\1 & 1 & {- 1} & {- 1} & {- 1} & {- 1} & 1 & 1 & 1 & 1 & {- 1} & {- 1} & {- 1} & {- 1} & 1 & 1 \\1 & {- 1} & 1 & {- 1} & {- 1} & 1 & {- 1} & 1 & 1 & {- 1} & 1 & {- 1} & {- 1} & 1 & {- 1} & 1 \\1 & 1 & 1 & 1 & {- 1} & {- 1} & {- 1} & {- 1} & {- 1} & {- 1} & {- 1} & {- 1} & 1 & 1 & 1 & 1 \\1 & {- 1} & {- 1} & 1 & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 & 1 & {- 1} & 1 & {- 1} & {- 1} & 1 \\1 & 1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 & 1 \\1 & {- 1} & 1 & {- 1} & 1 & {- 1} & 1 & {- 1} & {- 1} & 1 & {- 1} & 1 & {- 1} & {- 1} & {- 1} & 1 \\1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & {- 1} & {- 1} & {- 1} & {- 1} & {- 1} & 1 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 & {- 1} & 1 & 1 & {- 1} & {- 1} & {- 1} & 1 & {- 1} \\1 & 1 & {- 1} & {- 1} & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 & 1 & 1 & 1 & {- 1} & {- 1} \\1 & {- 1} & 1 & {- 1} & {- 1} & 1 & {- 1} & 1 & {- 1} & 1 & {- 1} & 1 & 1 & {- 1} & 1 & {- 1} \\1 & 1 & 1 & 1 & {- 1} & {- 1} & {- 1} & {- 1} & 1 & 1 & 1 & 1 & {- 1} & {- 1} & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1 & {- 1} & 1 & 1 & {- 1} & 1 & {- 1} & {- 1} & 1 & {- 1} & 1 & 1 & {- 1} \\1 & 1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} \\1 & {- 1} & 1 & {- 1} & 1 & {- 1} & 1 & {- 1} & 1 & {- 1} & 1 & {- 1} & 1 & {- 1} & 1 & {- 1}\end{bmatrix}$

[0029] Construction of orthogonal conversion matrix A:$C_{8} = \begin{bmatrix}16 & 0 & 0 & {- 2} & 5 & 0 & 0 & 2 \\0 & 16 & 2 & 0 & 0 & 5 & {- 2} & 0 \\0 & {- 2} & 16 & 0 & 0 & 2 & 5 & 0 \\2 & 0 & 0 & 16 & {- 2} & 0 & 0 & 5 \\{- 5} & 0 & 0 & 2 & 16 & 0 & 0 & 2 \\0 & {- 5} & {- 2} & 0 & 0 & 16 & {- 2} & 0 \\0 & 2 & {- 5} & 0 & 0 & 2 & 16 & 0 \\{- 2} & 0 & 0 & {- 5} & {- 2} & 0 & 0 & 16\end{bmatrix}$

[0030] In Matlab notation:

[0031] A=[zeros (8), zeros (8); zeros (8), C₈];

[0032] Tn=A*H16.

[0033] The values of the coefficients in the rows of the C8 matrix: 2,2, 5 and 16 in turn yield the square of the constant component 17, eachbeing squared and added up.

[0034] The odd base functions of T16 are derived as follows (again inMatlab notation):

[0035] T16 (2:2:16,:)=Tn ([9 13 11 15 10 14 12 16],:).

[0036] In other words, the 9th row of Tn becomes the 2nd row of T16, the13th row of Tn becomes the 4th row of T16, and so forth.

[0037] The advantage of this matrix T₁₆ is in addition to a highertransform gain the fact that it is constructed with the help of thehighly symmetrical matrices given above. Thus, efficient implementationin a coder and/or a corresponding decoder for transform coding of pixelinformation within blocks is possible.

[0038] The transform gain of an orthogonal transform having N basevectors is given by the ratio of the variance of the input signal afterquantization to the variance of the signal transformed with transformmatrix T_(N) and then quantized. This transform gain is usually given indB. In the literature, real video signals are frequently modeled by a1st order autoregressive process. This model is described completely bygiving the signal variance and the correlation coefficient. Assumingoptimum quantization and coding, the transform gain for this signalmodel may be given directly. For the correlation coefficient of 0.95,which is typical of video signals, a transform gain of 9.46 dB isobtained for DCT, 8.17 dB for the 16×16 matrix presented in the firstexample and 8.86 dB for the latter 16×16 matrix.

[0039] Coders and decoders for frequency range coding/decoding of movingimage sequences may be constructed with the method according to thepresent invention. Their transform devices must be designed so thattransform coefficients are processed in accordance with the method stepspresented above and/or the original moving image sequence isreconstructable from these coefficients.

[0040] For ICT matrices of the sizes 8×8 and 16×16, fast algorithms maybe developed which minimize the number of required additions andmultiplications.

[0041]FIG. 1 shows an example of such an algorithm schematicallyrepresenting the two transforms in a flow chart. ICT matrices of thesize 8×8 and 16×16 are marked by boxes labeled as T8 and T16 in FIG. 1.Since the even base functions of T16 correspond exactly to those of T8,the T8 block is completely integrated into the T18 block. Coefficientsof the input signal of length 16 are designated as x0, x1, x2, . . . ,xF and are on the left side of the chart. The coefficients of the outputsignal are 0y, 1y, 2y, . . . , yF.

[0042] Nodes in the graph represent additions. Multiplications byconstants are indicated by corresponding numbers at the edges. A minussign at one edge means subtraction instead of addition.

[0043] To make the diagram more comprehensible, the chart was divided atthe nodes marked by boxes, and a portion of the remaining course wasillustrated next to the main chart. The corresponding node points eachcarry the same designations.

[0044] The following table shows the number of additions andmultiplications required for the 8×8 transform and the 16×16 transform.For comparison purposes, the table contains information regarding thenumber of additions and multiplications for the fast Discrete CosineTransform according to [7]. T8 T16 Addition Multiplication AdditionMultiplication ICT 28 22 70 46 DCT acc. to [1] 24 16 72 44

[0045] The flow chart for fast implementation of the T16 and T8 matricesaccording to FIG. 1 contains some elements that occur repeatedly. Thisregularity reflects the symmetries within the transform. The chart issubdivided into four steps characterized by a perpendicular plane ofnode points. These are designated as steps 1 through 4 below.

[0046] In the first step, pairs of input coefficients are added andsubtracted, resulting in 16 coefficients again. The sums of x0 and xF,x1 and xE, x2 and xD, etc., form the input coefficients of the T8 blockwhich represents the even base functions. The differences are applied tonode 0-7 indicated in the figure. These then yield the odd components.The star structure resulting from these additions and subtractions isrepeated with 8 coefficients instead of 16 at the input of the blockdesignated as T8 in step 2; then with the four upper coefficients (evenbase functions of T8) at step 3 and with two coefficients before outputsy0 and yF at step 4. This structure is equivalent to fast implementationof the Discrete Cosine Transform.

[0047] Stars that do not represent pure additions and subtractions butinstead contain weighting coefficients also occur repeatedly. Thus, twoweighted stars with four coefficients are removed from nodes 3-4, 2-5,1-6 and 0-7 indicated in FIG. 1; these differ only in the arrangement ofweights. This structure is repeated before outputs y4 and yC with otherweights and two coefficients.

[0048] For the odd base functions of T16, there are variations of thisweighted structure. At step 3 there are four structures here, eachhaving two outputs with weighted coefficients and two outputs with pureadditions/summations. These differ in the distribution of weights. Inthree of these structures, only two nodes form the input, so thisresults in distorted stars.

[0049] Literature

[0050] [1] Telecom. Standardization Sector of ITU, “New integertransforms for H.26L”, in Study Group 16, Question 15, Meeting J,(Osaka, Japan), ITU, May 2000.

[0051] [2] Telecom. Standardization Sector of ITU, “Addition of 8×8transform to H.26L”, in Study Group 16, Question 15, Meeting I, (RedBank, New Jersey), ITU, October 1999.

[0052] [3] W. Cham, “Development of integer cosine transforms by theprinciple of dyadic symmetry”, IEE Proc., vol. 136, pp. 276-282, August1999.

[0053] [4] W. Cham and Y. Chan, “An order-16 integer cosine transform,”IEEE Trans. Signal Processing, vol.39, pp.1205 1208, May 1991.

[0054] [5] R. Srinivasan and K. Rao, “An approximation to the discretecosine transform for n=16”, Signal Processing 5, pp. 81-85, 1983.

[0055] [6] A. K. Jain, Fundamentals of digital image processing.Englewood Cliffs, N.J.: Prentice Hall, 1989.

[0056] [7] W. H. Chen, C. H. Smith, and S. C. Fralick, “A FastComputatial Algorithm for the Discrete Cosine Transform”, IEEE Trans.Comm., Vol. COM-25, No. 9, September 1977, pp. 1004 1009,

What is claimed is:
 1. A method of obtaining approximated integer cosine transform coefficients, in particular for coding pixel blocks, the transform coefficients being selected as follows: a limited value range is defined for the transform coefficients, the coefficients of the base vectors for the submatrices are selected, taking into account the orthogonality condition, so that the sum of their squares yields the square of the constant component coefficient, the coefficients of the variable components are derived from these coefficients.
 2. The method as recited in claim 1, wherein a uniform normalization factor and/or quantization factor is used for all coefficients.
 3. The method as recited in claim 1 or 2, wherein block sizes coupled to the movement compensation are used for the transform.
 4. The method as recited in one of claims 1 through 3, wherein the integer cosine transform matrices are generated with the aid of a conversion matrix from the Hadamard transform of the same matrix size, for example, 16×16.
 5. The method as recited in one of claims 1 through 4, wherein the value 17 is selected for the coefficients of the base vectors for the constant component.
 6. The method as recited in claim 4 and 5, wherein the coefficients of the base vectors for the lowest variable component are selected to have absolute values of 15 and
 8. 7. The method as recited in claims 4 and 5 or 6, wherein the coefficients of the base vectors for the second lowest variable component are selected to have absolute values of 12, 9 and
 8. 8. The method as recited in claims 4 and 5 or 6 or 7, wherein the coefficients of the base vectors for the third lowest variable component are selected to have absolute values of 13, 10, 4 and
 2. 9. The method as recited in claims 4 and 5 or 6 or 7, wherein the coefficients of the base vectors for the third lowest variable component are selected to have absolute values of 16, 5, 2 and
 2. 10. A coder for frequency range coding of moving image sequences, having a transform device which is designed to create transform coefficients for a moving image sequence, these coefficients being processed by the method steps according to claims 1 through
 9. 11. A decoder for frequency range decoding of moving image sequences, having a transform device designed to reconstruct a moving image sequence from transform coefficients created by using the method steps according to claims 1 through
 9. 12. The method as recited in one of claims 1 through 9, wherein essentially symmetrical algorithms are used for the approximated transforms, the input-end transform coefficients being supplied, in each case, step by step to an addition node and/or a subtraction node and weighted accordingly for the required multiplication operations. 